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Full text of "Thermodynamics, Thermal Radiation and the Beginnings of Modern Physics"

Thermodynamic s, Thsnr.pl Radiation and the Beginning* 



or 



luantum Phvsics, 



P. 3.0. Bruskiewich 
Mathematical-Physics, University ©f British Columbia, Vancouver B.C. 

The beginnings of Quantum Theory is introduced in such 
a manner as to present the continuity of effort between the 
"classical" and "modern" physics. Emphasis will be on the 
specific problem of the energy spectrum of a blackbody source 
in thermal equilibrium, and the transfer of energy on the atom- 
ic scale. 



In the year 1900»Max Planck 
presented a paper to the Grerras.n 
Physical Society in Berlin(l) that 
helped to change the intrinsicJchara- 
cter of science, Eis concern m this 
paper was with the' mechanical ,thermo 
dynamic and elect rodynamic processes 
specifically associated with " ideal" 
resonators inside an "ideal" black- 
body enclosure. In a sense, his work 
represented the consistent extension 
of the endeavours of many other"clasa 
ical" physicists like Kirchoff ,Clausius > 
Maxwell, St ©fan ,Wi en and Bolt smarm, but 
it also represented a personal daring 
(encompassed in the introduction of the 
concept of discrete action ) that was 
later to blossom into what we now call 
quantum physics, Needless to say, at the 
time his approach did not follow that 
of others ( James Jeans, for instance), 
but the conclusions, though originally 
heuristic and highly theoretical, that 
can be drawn from his work have shown, 
beyond any doubt, that Planck's concept 
of discrete action is very fundamental 
to the intrinsic understanding of the 
physical universe. 

There are many ways one could 
approach Planck's achievement, however, 
to appreciate the fundamental importance 
of his work it would be best to follow 
Planck' s own formulation of his blackbody 
spectral law. 

Max Planck' s career as a physic 
ist began with over fifteen years ©f 
study in the field of thermodynamics, mere 
specifically the second law, thermodynamic 
potentiality, and irreversibility, Because 
of his beginnings (as a diadple of Rudolf 



Olauaius, the formulator of the 
second law ) , Planck considered 
the second lav; to be universally 
valid, namely processes in which 
the total entrop«> f a system decreg 
sed over extended time were to be 
considered srictly impossible. '. 
originally did not consider Ludwig 
Boltsmann' s reformulation of the 
second lav/ (into a statistical law) 
as being acceptable because Boltzmann's 
statistical mechanics did not make - 
the increase of entropy absolutely 
certain, only highly probable. Planck 
often expressed the deep, esoteric 
conviction that the loftiest goal 
of any if not all science was the 
search for the absolute.This convict 
ion is clearly reflected in his 
appreciation of the general physical 
laws, but beyond the formulation of 
fundamental laws, he attempted, as 
others attempted before him (e.g. 
Ludwig Boltzmann ) , to relate the 
unexplained physical processes and 
paradoxes of his time, to fundament 
ally universal laws.Flanck believed 
that the Principle of Increasing 
Entropy could bo preserved intack as 
a rigorous theorem in a comprehensive 
world view .When it came to the unexp 
lained phenomenon of blackbody radia 
tion, it was this belief that acted 
as the touchstone to Planck's attempt 
at expl aination* 

In March I895 Planck presented 
a paper(2) in which he discussed the 
problem of resonant scattering by an 
oscillating dlpole(of dimension comp 
atible with the wavelength) , of a 



pi 



" 



plane elctromagnetic wave(5) .'What this 
scattering process offered to him was a 
way of relating the equilibrium ctate(4) 
radiation in a blackbody enclosure to 
the states of hypothetically ideal re son 
at or s } which Planck introduced as making 
up the walls of an ideal blackbody enclo 
sure(5) • Planck studied the mechanisms 
of emi s si on/ab sorption end of damping of 
these hypothetical resonators ;He realised 
that by disregarding mechanical damping 
by concerning himself uniquely with radi 
ation damping, and by taking the mechanism 
for the system's irreversibility as being 
the conversion of the incident plane waves 
into spherical waves, you could more fully 
begin to understand the dynamics of the 
radiation equilibrium in the enclosure. 
However, a very fundamental flaw in his 
approach was brought up by Boltzmann(it 
was a flaw that forced Planck to change 
his attitude towards Boltzmann 1 s statist! 
cal mechanics ) • The equations of electro 
dynamics do not lead to a monotonic conver 
gence to a state of thermal equilibrium, 
Nothing in the laws of electrodynamics 
prevent the inverse of the scattering 
processes from occuring^This important 
fact forced Planck to admit that statist! 
cal descriptions were necessary(6).It also 
forced him to consider energy exchange by 
.discrete means as opposed to continuous 
energy exchange. 

Backtracking alittle bit, we can 
now begin to approach the mathematics of 
Planck's concept. Experiments by Josef 
Stefan in 1879 j s -nd theoretical work by 
Ludwig Boltzmann in 1884, established that 



JD(1T.- 



law. Wien* a spectral law was 
be a reasonable representation of the 
blackbody spectral distribution known 
up to 1899* In 1899 > Planck, continuing 
hi 3 work begun in 1895 on ^n e analogy 
of the" ideal" resonators, proved (7) 
that the distribution function must be 
of the form 

where 3(v,T) 
represents the average energy of the 
"ideal" resonators at frequency v, at 
cavity temperature. Though this equati 
on applies to a hypothetical wall comp 
osed of "ideal" resonators, because of 
Kirchoff ' s 1&59 proof regarding the 
constituents of the walls of a black* 
body enclosure, once a valid frequency- 
temperature relation is found for the 
average resonator energy E(v»T), it can 
be applied to a cavity formed by any 
real material. Because of" his belief 
in the universality of the second law* 
Planck attempted to discern the form 
of 3(v,T) by looking at the fundamental 
relationship between the energy and 
entropy of a system in a state of thermal 
equilibrium. He began with 



dfc 



T 



, and by 

differentiating again he came up with 
the equally important equation 






J — 



„' proved rigorously that the distribution 
function p(v,T) must be of the form 



The derivative dS/dT 
could be evaluated by combining Wien 1 s 
spectral law with Planck's distribution 
In 189 2 ) Wilhelm Wien function so that « „, 

<trt* ' 






p(at,t) - \r S-iy/r) 






In I896, ";;ien, using 
a somewhat questionable theoretical present 
ation, proposed that f(v,T) was of the form 



U«/t) " A £ 



«>r 



1 



. 



anc 



The entropy - energy derivative 
then becomes 




ent constants, giving 



p(v,T} ■= Av^e, * 



where A & B rep res 



Rv 






|5irE 



The simplicity of this entropy- 
energy derivative greatly impressed 
Planck, but spectral measurements taken 
Wien 1 s spectral (at the same time that Planck did his work 



(2) 



in 1899) &t high temperatures and low 
frequencies mads it apparent that T >fien' s 
spectral lav; had serious empirical limit 
ations. In 1900 , working with quite new 
data ^Planck derived a simple dependence, 
which agreed with the high temperature 
data (i.e. p(v,T)c*iT, 2(v,T><r, dE/dT is 
constant ) .Planck realized that these two 
\* <js cases were limiting cases, one for relati 
VS^y?" ^ high energy and the other for low. 
' $-* ' Almost within the same few days he formul 
o, ' ■ atc-a a single equation which sufficed to 
"' X 6 «- exoress both limiting cases, namely 



« -. - \e (a +& )} 



-I 



this single equation yields 



Integration of 



41 




CA-tG)] 4-i 



with d serving 
as the constant of integrstion.Replacing 
dS/dE by l/T, the equation becomes 






is\ 



e 



A 



a — rr; 



"valuation of 
the constant of integration(i.e. looking 
at the equation as l/T •> ) gives d - 0. 
Solving the last equation yielded an appr 
opriate equation for the resonant energy, 
namely 



(l%~ 



giving the expression 






A 



r 



3 



(*>~0 



and p(v,T) the form 



Q-v 



Ji 



P<W7 = 7—^ 



which can be evaluated 
for B and thenceforth G, giving 



" ~F 



av» 



/4 ^ - SjLvfk 

h being a new universal constant now 
known as Planck's constant (h-6. 6?,6 \10"^ T 
J-sec). 

Planck, himself a theoretician, 
was not satisfied with just fitting a 
formulation to empirical data and began 
to put his concepts into a rigorous 
form. As I pointed out before, criticism 
By Ludwig Boltzmann forced Planck to 
turn towards a statistic;! interpretation 
of entropy .This statistical approach 
yielded dramatic results. 

Planck began by looking at the N 
ideal" resonators making up the walls of 
his hypothetical blackbody enclosure .Each 
resonator has an average energy<E>and an 
average entropy^S^ giving a total energy 
E t of 11(E) and a total entropy S^of l$>.It 
is here that Flanck introduced Boltzmann' s 
famous expression pfL ^i^-wv-a^wtl ^<J\j^>Jn{(J3\ 

n«Hw»l ,and follow&% 

the^Bolztmann devised previously to deal 
with the entropy divided amongst many 
oscillators(8) . By proposing that the total 
energy E. be made up of n discrete units 
of energy, each of amount e,he was able 
to equate 1^2) to ne and open up the approach 
to the fundamental problem of W; How to 
calculate the number of ways n discrete 



units of energy can- 



distributed amongst 



for the distribu 
tion function. Returning to the limiting 
case encompassed in Wien' s spectral law, 
Planck showed that A, a (supposive) constant* 
really was a function depending on v, so 
that A = F' v% with P representing a(real } 
constant .Thus f(v,T) takes on the form 



N resonators. More fundamentally, Planck 
realized that previous limitations in his 
reasoning, as well as in ioltzmann's work, 
could be removed by considering the energy 
transfer to actually be occuring by discrete 
means. 

Flanck now had a way of relating 
W to N and n. Beginning with 



\r 



I 






-+K -0 



lAt (ti-i 



and making the 



(5) 



approximation { on the assumption that 
both N and. n are largo ) 



expression 



.•■„ 



■v\)l 



M- 



and by using 
Stirling" s ftyyiitU - l a a-j^^/je/Ci ha <*wk_ 



y\ 



-ti\ ~ ~t{v\-L--t 



he wag able to 



express klnW as 



jUfi-fhe entropy of 
single "ideal" resonator is 



a 



s - 



c' 



He transformed this 
entropy relation into an energy 
relation ££r the taking jj^the 
derivative 



X> - 



d£ \e 



•\ 



l ? A 



E 



and by (again) equating 
this to 1/T, so then 



j(/\ 



e, v 



G 



\ 



e 



Solving for 2(v,T), he 
ended up with his previously derived 
expression 



eKt 



q, " x 



By returning to Wien's 
distribution function 



PKtV nrHOf 



and his own 



he - was ablo- te reasoned 
fl^fe- that e«w or (again) returning to Planck 1 ' a 
prf ri i OTn fnrrmi 3 m ti l <m J thfiti" 5 hv. Thus 



he was able t 
function 



1 1 . * ' ■*■ 



his distribution 







(a 5 * -i; 

quite rigorously. 
TMsenew radiation formulae 
brought a great deal of pleasure^' rSei. 
they were in agreement with the then new 
spectral data of Kurlbsum and Rubens, who 
at the time were doing precise spectral 
measurements at high temperature and low 
frequencies ) , but also a great deal of 
distress in that, if one accepte d his 
formulation^ th on"^he.rgy„tran sfer appear ed^ 
to be occuring b y discrete steps as opposed 
to the cTxLssicaliBSEIc] continuous transfer , 

Planck, even though he was 
the formulator of this important discovery, 
refused for many years, to accept fully 

the consequences of discrete energy transfers 
It was left to someone as bold and ira-gina 
tive as Albert Sinstien to expand Planck's 

original work -IwtJfW luiU maim^ JL ^Mx/SW^ ^ 

************ pLfiiw, 

(1) Zur Theorie des Gesetzes der 3ner<tie~ 
vertoilung in Normalspecrtu m, M.Planck, Verb. 
d.Deutsch.Phys.Ges. 2, 202,(1900) , ( english 
translation! Planck's Original Papers ,Taylor 
and Francis, 1972) ., 

(2)M.Planck,Ann.Hrys.(5) 57,1,(1396) . 
(5) The intensity of the plane polarised light 
is independent of the orientation of the plane 
of polarization as well as the position and 
direction of the light. Inside the enclosure 
the radiation is isotropic and homogeneous. 
(4) The state of thermodynamic equilibrium 
is that state in which the entropy of the 
system has the maximum value compatible with 
the total energy as fixed by the initial 
conditions of the system, 
(5) Planck was able to use the analogy of 
discrete "ideal" resonators as constituting 
the enclosure's walls because Gustav Kirchoff 
had previously provelft (in 1©?9) that the 
energy spectrum of a large blackbody enclosure 
is entirely independent of the constituents, 
shapes and sizes of the materials making up 
its walls. 



(4) 



(6) In thia resolution lay dormant the 
complete theoretical description of the 
mechanism of radiation. 

(7) How Planck aid this is best seen in 
the englieh translation of his book 
Waermestrahlung (191?) » by M.Masiua, 
Blakiston's Son & Qo, 19l4 f titled 
Planck's Heat P.adiation « 

(8) Boltzmann originally introduced this 
statistical description as a computational 
aid, however ; later he became quite satisfied 
to consider its application as being more 



iAott~ w Mew 



general. 






fc 



f^^A >(**<- 






L l P 



. fc° 



Dfl 



Qoihf*** 









Kii^ , FHp'i6Q Tcte^ ^t* resell